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Galois Representations and Diophantine Problems

Wiles' proof of Fermat's Last Theorem was one of the crowningachievements of 20th century mathematics, and has led to twomajor research programmes. The first aims to establish moregeneral and powerful modularity theorems for Galois representations,and has yielded the proof of modularity of elliptic curves over therationals by Wiles, Breuil, Conrad, Diamond and Taylor, the proof ofSerre's modularity conjecture by Khare and Wintenberger, and powerfulmodularity lifting theorems by Kisin, Barnet-Lamb, Gee, Geraghty andothers. The second programme aims to apply the proof strategy ofFermat's Last Theorem to solve Diophantine problems, especially casesof the notorious Beal conjecture (also known as the generalized Fermat conjecture).The original strategy of Hellegouarch, Frey, Serre andRibet has been refined and extended by many, most notably Bennett,Dahmen, Darmon and Kraus. The two programmes are still intricatelyconnected, as evident in the recent work of Freitas and Siksek on theFermat equation over totally real fields.

Syllabus:

The minicourse will use Diophantine equations as a vehicle tointroduce and motivate the study of Galois representations of ellipticcurves and of modular forms and their relationship (modularity). Thestudents will learn the basics of 2-dimensional Galoisrepresentations, understand the statements of modularity theorems(including the more recent modularity lifting theorems), and how toapply them to solve basic ternary Diophantine problems.